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On an inverse problem in electromagnetism with local data: stability and uniqueness
A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration
1. | Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, United Kingdom |
2. | Institute of Biomathematics and Biometry, Helmholtz Zentrum München, Ingolstädter Landstrasse 1, 85764 Neuherberg, Germany |
3. | Department of Mathematics, Humboldt-University of Berlin, Unter den Linden 6, 10099 Berlin, Germany |
References:
[1] |
USC-SIPI image database, In A. Weber, editor,, , ().
|
[2] |
A. Bovik, "Handbook of Image and Video Processing," Academic Press, 2000. |
[3] |
A. Brandt, Guide to multigrid development, In "Multigrid Methods" (Cologne, 1981), volume 960 of Lecture Notes in Math., pages 220-312. Springer, 1982. |
[4] |
A. Chambolle, An algorithm for total variation minimization and application, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97. |
[5] |
A. Chambolle and P-L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188.
doi: 10.1007/s002110050258. |
[6] |
T. Chan, K. Chen and J. L. Carter, Iterative methods for solving the dual formulation arising from image restoration, Electronic Transactions on Numerical Analysis, 26 (2007), 299-311. |
[7] |
T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.
doi: 10.1137/S1064827596299767. |
[8] |
Q. Chang and I-L. Chern, Acceleration methods for total variation-based image denoising, SIAM J. Applied Mathematics, 25 (2003), 982-994. |
[9] |
K. Chen, "Matrix Preconditioning Techniques and Applications," Cambridge University Press, 2005.
doi: 10.1017/CBO9780511543258. |
[10] |
D. C. Dobson and C. R. Vogel, Convergence of an iterative method for total variation denoising, SIAM J. Numer. Anal., 34 (1997), 1779-1791.
doi: 10.1137/S003614299528701X. |
[11] |
I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," Classics Appl. Math. 28, SIAM, Philadelphia, 1999. |
[12] |
C. Frohn-Schauf, S. Henn and K. Witsch, Nonlinear multigrid methods for total variation image denoising, Comput. Vis. Sci., 7 (2004), 199-206. |
[13] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Birkhäuser, Boston, 1984. |
[14] |
W. Hackbusch, "Multigrid Methods and Applications," volume 4 of Springer Series in Computational Mathematics, Springer-Verlag, 1985. |
[15] |
M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth newton method, SIAM J. Optim., 13 (2002), 865-888.
doi: 10.1137/S1052623401383558. |
[16] |
M. Hintermüller and K. Kunisch, Total bounded variation regularization as bilaterally constrained optimization problem, SIAM J. Appl. Math., 64 (2004), 1311-1333.
doi: 10.1137/S0036139903422784. |
[17] |
M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM Journal on Scientific Computing, 28 (2006), 1-23.
doi: 10.1137/040613263. |
[18] |
M. Hintermüller and M. Ulbrich, A mesh-independence result for semismooth Newton methods, Math. Program., 101 (2004), 151-184. |
[19] |
P. J. Huber, Robust regression: Asymptotics, conjectures, and Monte Carlo, Annals of Statistics, 1 (1973), 799-821.
doi: 10.1214/aos/1176342503. |
[20] |
S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, SIAM Multiscale Model. and Simu., 4 (2005), 460-489.
doi: 10.1137/040605412. |
[21] |
L. Rudin, "MTV-Multiscale Total Variation Principle for a PDE-Based Solution to Nonsmooth Ill-Posed Problem," Technical report, Cognitech, Inc. Talk presented at the Workshop on Mathematical Methods in Computer Vision, University of Minnesota, 1995. |
[22] |
L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[23] |
D. Strong and T. Chan, "Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing," Technical report, UCLA, 1996. |
[24] |
D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 19 (2003), 165-187.
doi: 10.1088/0266-5611/19/6/059. |
[25] |
X.-C. Tai, "Rate of Convergence for Some Constraint Decomposition Methods for Nonlinear Variational Inequalities," Numerische Mathematik, 2003. |
[26] |
X.-C. Tai, B. Heimsund and J. Xu, Rate of convergence for parallel subspace correction methods for nonlinear variational inequalities, In "Thirteenth International Domain Decomposition Conference" (Barcelona, Spain, 2002). |
[27] |
U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid," Academic Press, 2001. |
[28] |
P. S. Vassilevski and J. G. Wade, A comparison of multilevel methods for total variation regularization, Electron. Trans. Numer. Anal., 6 (1997), 255-270. Special issue on multilevel methods (Copper Mountain, CO, 1997). |
[29] |
C. R. Vogel, A multigrid method for total variation-based image denoising, In "Computation and Control, IV" (Bozeman, MT, 1994), volume 20 of Progr. Systems Control Theory, pages 323-331. Birkhauser Boston, 1995. |
[30] |
C. R. Vogel and M. E. Oman, "Iterative Methods for Total Variation Denoising," SIAM Journal on Scientific Computing, 1996. |
[31] |
C. R. Vogel, "Computational Methods for Inverse Problems," Frontiers Appl. Math. 23, SIAM Philadelphia, 2002. |
[32] |
P. Wesseling, "An Introduction to Multigrid Methods," John Wiley and Sons, 1992. |
show all references
References:
[1] |
USC-SIPI image database, In A. Weber, editor,, , ().
|
[2] |
A. Bovik, "Handbook of Image and Video Processing," Academic Press, 2000. |
[3] |
A. Brandt, Guide to multigrid development, In "Multigrid Methods" (Cologne, 1981), volume 960 of Lecture Notes in Math., pages 220-312. Springer, 1982. |
[4] |
A. Chambolle, An algorithm for total variation minimization and application, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97. |
[5] |
A. Chambolle and P-L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188.
doi: 10.1007/s002110050258. |
[6] |
T. Chan, K. Chen and J. L. Carter, Iterative methods for solving the dual formulation arising from image restoration, Electronic Transactions on Numerical Analysis, 26 (2007), 299-311. |
[7] |
T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.
doi: 10.1137/S1064827596299767. |
[8] |
Q. Chang and I-L. Chern, Acceleration methods for total variation-based image denoising, SIAM J. Applied Mathematics, 25 (2003), 982-994. |
[9] |
K. Chen, "Matrix Preconditioning Techniques and Applications," Cambridge University Press, 2005.
doi: 10.1017/CBO9780511543258. |
[10] |
D. C. Dobson and C. R. Vogel, Convergence of an iterative method for total variation denoising, SIAM J. Numer. Anal., 34 (1997), 1779-1791.
doi: 10.1137/S003614299528701X. |
[11] |
I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," Classics Appl. Math. 28, SIAM, Philadelphia, 1999. |
[12] |
C. Frohn-Schauf, S. Henn and K. Witsch, Nonlinear multigrid methods for total variation image denoising, Comput. Vis. Sci., 7 (2004), 199-206. |
[13] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Birkhäuser, Boston, 1984. |
[14] |
W. Hackbusch, "Multigrid Methods and Applications," volume 4 of Springer Series in Computational Mathematics, Springer-Verlag, 1985. |
[15] |
M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth newton method, SIAM J. Optim., 13 (2002), 865-888.
doi: 10.1137/S1052623401383558. |
[16] |
M. Hintermüller and K. Kunisch, Total bounded variation regularization as bilaterally constrained optimization problem, SIAM J. Appl. Math., 64 (2004), 1311-1333.
doi: 10.1137/S0036139903422784. |
[17] |
M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM Journal on Scientific Computing, 28 (2006), 1-23.
doi: 10.1137/040613263. |
[18] |
M. Hintermüller and M. Ulbrich, A mesh-independence result for semismooth Newton methods, Math. Program., 101 (2004), 151-184. |
[19] |
P. J. Huber, Robust regression: Asymptotics, conjectures, and Monte Carlo, Annals of Statistics, 1 (1973), 799-821.
doi: 10.1214/aos/1176342503. |
[20] |
S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, SIAM Multiscale Model. and Simu., 4 (2005), 460-489.
doi: 10.1137/040605412. |
[21] |
L. Rudin, "MTV-Multiscale Total Variation Principle for a PDE-Based Solution to Nonsmooth Ill-Posed Problem," Technical report, Cognitech, Inc. Talk presented at the Workshop on Mathematical Methods in Computer Vision, University of Minnesota, 1995. |
[22] |
L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[23] |
D. Strong and T. Chan, "Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing," Technical report, UCLA, 1996. |
[24] |
D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 19 (2003), 165-187.
doi: 10.1088/0266-5611/19/6/059. |
[25] |
X.-C. Tai, "Rate of Convergence for Some Constraint Decomposition Methods for Nonlinear Variational Inequalities," Numerische Mathematik, 2003. |
[26] |
X.-C. Tai, B. Heimsund and J. Xu, Rate of convergence for parallel subspace correction methods for nonlinear variational inequalities, In "Thirteenth International Domain Decomposition Conference" (Barcelona, Spain, 2002). |
[27] |
U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid," Academic Press, 2001. |
[28] |
P. S. Vassilevski and J. G. Wade, A comparison of multilevel methods for total variation regularization, Electron. Trans. Numer. Anal., 6 (1997), 255-270. Special issue on multilevel methods (Copper Mountain, CO, 1997). |
[29] |
C. R. Vogel, A multigrid method for total variation-based image denoising, In "Computation and Control, IV" (Bozeman, MT, 1994), volume 20 of Progr. Systems Control Theory, pages 323-331. Birkhauser Boston, 1995. |
[30] |
C. R. Vogel and M. E. Oman, "Iterative Methods for Total Variation Denoising," SIAM Journal on Scientific Computing, 1996. |
[31] |
C. R. Vogel, "Computational Methods for Inverse Problems," Frontiers Appl. Math. 23, SIAM Philadelphia, 2002. |
[32] |
P. Wesseling, "An Introduction to Multigrid Methods," John Wiley and Sons, 1992. |
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